First off, the below is mostly assuming that you’re going to have a conversation over the phone about the proposed salary and such in your offer letter. If the idea of doing this over the phone really, really frightens you, though, there is no shame in conducting it in another format that’s less intimidating to you, like email. You should do whatever you feel will set you up the best for succeeding in getting what you ask for, while noting that it tends to be easier to adapt to the atmosphere over the phone than through a "cold" medium like email, where it's easier for people to assume the worst.
And now, some thoughts on how to think about the whole negotiation process:
If this is your first salary negotiation, I think it helps to somewhat set a goal that feels really achievable. For me, it was just to go through with trying to negotiate it my salary at all. I felt like I was going to puke all day beforehand, I was so nervous--but hey, I didn’t die! So next time has to be better than that!
One perspective that made the salary negotiation process feel a lot more palatable to me is to NOT think of it an adversarial situation but instead, to think about it as both you and who you’re negotiating with have a shared goal of working together to make this offer even better for both you and the company. You don’t at all have to lie or pretend like you’re not interested, if those are tactics that make you feel uncomfortable (they do for me, for sure). Instead, you can make the conversation sound really positive and express how excited you are about the opportunity, they're excited, and an attitude of “let's work together to make it even better!” The official offer is not a binding contract, it’s just a starting point for your conversation.
I also prefer to think of this as NOT a reflection or judgement on your value as a person! I’m not a fan of advice that’s centered on “asking for what you’re worth” because that’s tied to your own, potentially wrong (in either too positive or too negative) assessment of what you’re “worth.” Instead, it should be all about the value you bring to the company and how hard it would be for them to find someone to replace you. This is also just a much more business-savvy way to go about the negotiation, where you’re not asking for more just because you’re a special snowflake, but because you’re a special snowflake that they are investing in and will get an excellent return on in value you provide.
If you are still nervous about the whole prospect of negotiating your salary in light of the story of the philosophy professor whose offer was rescinded after negotiating over email, this blog post on How Not to Negotiate Your Next Job is a good read. I particularly like #4 and #6 on her list of guidelines.
Next up, I’ll get into some details on how to figure out what exactly what you want to ask for, which is probably going to end up being the longest post in this series.
Update: here's the full series of posts